Supersymmetric models are closely associated to complex geometry in several different ways.
To be able to write some extended models in a manifest form superspaces may be extended with a \(CP^1\)  at each point thus relating them to twistors [1]}, conformal supergravity can be formulated in terms of local twistors [2]}, [3]}, and supersymmetic nonlinear sigma models is often best formulated in terms of complex superfields and typically have complex target space geometries [4]},[5]}. This last property is what shall concern us here, although we have to mainly restrict to two-dimensional models with two left and two right going supersymmetries i,e, to \(2d\) , \((2,2)\)  supersymmetry. Fortunately this case is sufficiently rich to warrant independent scrutiny.
